Armature not staying in place
I think that you first need to make some armature corrections :


Rotate your IK controller in Edit mode to make it more easy to handle.
Deactivate its Deform option as it is not supposed to influence the mesh.
Unparent it, it is not supposed to be child of the hand, it is supposed to be parented with offset to the root bone.
Select the hand and make it the child of the controller.
You may also need to move the Pole Target, and also deactivate Deform.


Now when you select the mesh in Weight Paint mode, you can see that the chest bone influences the arms while it is not supposed to, so use the brush to correct that.